is MSD of the random walker. Despite the mean displacement, MSD is non-zero quantity and is defined in the sense of statistical average as: hx2(t)i=h(PNi=1si)2i.
Rewriting the sum one gets hx2(t)i=h(
The second sum term in the expression above is zero due to independence of the jumps, namely absence of any correlation. Calling the assumptions that the second moment of jumps is finite, and in average is< si2>=a2, and the step time intervals are equal, he reached the MSD to be
< x2(t)>= 2Dt (1.1)
where D is the diffusion coefficient being D= 2τa2
0. Based on these assumptions, Einstein could derive the differential equation for PDF of tracer’s displacement
∂
∂tP(x, t) =D ∂2
∂2xP(x, t) (1.2)
which its solutions is a Gaussian function, P(x, t) = √ 1
4πDtexp(−x4Dt2), satisfying the initial condition ofP(x,0) =δ(x). The same diffusion equation was first introduced by Fick (?), in 1985 following a phenomenological approach. Combining the linear response theory for concentration of diffusing particles and conservation of number of particles, Fick derived the same formula as of Einstein with simple replacement of number of particles with PDF of tracer.
Therefore, normal diffusion is characterized with the linear growth of MSD with time lag and Gaussian form of the PDF ofthe particle’s displacement. Additionally, the PDF has a scaling form such that, higher moments of displacement can be written as a function of second moment. Therefore, knowing the MSD for a normal diffusion is sufficient to calculate the PDF and higher moments of the particle’s displacement. Any deviation from these properties in diffusion process belongs to another class of diffusion, termed anomalous diffusion, which we will discuss in the following section.
1.3 Anomalous Diffusion
In Einstein’s approach to the problem of diffusion, he made assumptions which are necessary not the case for diffusion of the particle in complex and heterogeneous sys-tems such as biological ones. Plethora of observations unveiled different behaviours of the MSD and PDF of the particle’s displacement compared to the predictions of Einstein’s theory. The MSD of the particle in such observations did not follow
1 Introduction
the linear dependence on time and showed a rather complicated non-linear depen-dence as: hx(t)2i = 2Dtα withα 6= 1 being exponent of anomalous diffusion. The PDF of the particle’s displacement, however, happened to be Gaussian or deviate from Gaussian form for different observations with the non-linear time dependency of the MSD. These deviations persist for time scales much larger than characteris-tics diffusion time and cease to exist in times which could be longer than a specific measurement time.
Those processes with exponent of anomalous diffusion greater than unity,α >1, are assigned to the class of super-diffusion. Such situations occur when the second moment of step length PDF does not remain finite. The other corresponding situa-tion is when the steps in random walk are positively correlated such that the walk is persistent. As examples to such situations, one could refer to the random walk on the chemical space of polymer chains and transport of particle (protein) made by molecular motors(???).
The other class of anomalous diffusion which is the main subject of this work, is called sub-diffusion. Sub-diffusion refers to the case that the exponent of anomalous diffusion is smaller than unity, α < 1. Diffusion process in this situation is much slower than normal one.
First physical scenario leading to sub-diffusion is when the mean waiting time to perform a walk is much longer than the observation time and in this sense di-verges. This situation was first assumed to describe the transport process of charge carriers in amorphous material(?). In biological sense, one can imagine a trapping-untrapping effect in binding sites, for example in living cell. The mathematical model for such physical assumption is called continuous time Random Walk model (CTRW). In CTRW model, not only the MSD may shows a non-linear dependency on time lag but also the PDF of the particle’s displacement sharply deviates from Gaussian.
The other scenario for sub-diffusion is when the displacements are anti-correlated.
This means in the sense that in jump action, there is higher probability that a particle moves in the direction opposite to the last one. This anti-correlation of displacement may be imposed by a structural disorder in the media in which diffusion takes place.
The model for such a situation generally referred as random walk on fractal. The PDF in this model is also non-Gaussian, however, the non-Gaussianity is much smaller then what is observed in CTRW model.
Another possibility for the anti-correlation of steps is diffusion in visco-elastic environments. In such cases, different parts of the system create an interacting complex and the particle has to move through a concreted way. The corresponding theoretical model is generalized Langevin equation (GLE) and the fractional Brow-nian motion (fBm) which is closely related to (GLE). Contrary to the other models of sub-diffusion, these models show a Gaussian PDF of the particle’s displacement.
In what follows, we shortly discuss the mentioned mathematical models for the
6
1.3 Anomalous Diffusion
sub-diffusion. Understanding the fundamental properties and differences between such models will be utilized to recognize them using fluorescence spectroscopy tech-nique.
1.3.1 Continuous Time Random Walk
Continuous time random walk (CTRW) is one of the most popular methods to de-scribe the sub-diffusion and is widely discussed in the recent book (?). In CTRW, the medium in which diffusion takes place is considered as a homogeneous space containing randomly distributed traps. Despite to the Einstein’s picture of nor-mal diffusion, sojourn time of the particle in these traps is not identical and varies depending on the depth of energetic traps encoded in waiting time probability dis-tribution function,ψ(t), from which the waiting time for next jump is drawn. If this probability density functions have the first moment, the mean sojourn time is finite , τ0 = R0∞tψ(t)dt . Then the average number of steps in time t is hn(t)i= τt
0 and
this results in normal diffusion.
For those waiting time probability density functions that lack the first moment, for example heavy tailed power law distribution function ψ(τ) ∝ τ−1−α with 0 <
α <1, the number of steps goes ashn(t)i= (tt
c)α withtc being some characteristic time. The jump length of the particle is, however, given from the PDF of the jump length, p(x). Assuming that the PDF of the jump length has the second moment in accordance to Einstein’s assumption,a2=R−∞+∞x2p(x)dx, the MSD of walker for sub-diffusion is then given as
hx(t)2i ∝ a2
tαc tα. (1.3)
To obtain the average number of steps at time t, hn(t)i = P∞0 nχn(t), one may note the , χn, being the probability of performing exactly n steps by the time t. Probability of making no step is given by, χ0 = R0∞ψ(t0)dt0, and other χn are calculated iteratively asχn=R0∞ψ(t0)χn−1(t−t0)dt0. Passing to the Laplace domain, one can obtain the correspondence of the internal clocknof CTRW and the physical timet given byχn(t) in the Laplace domain as
χn(s) =ψn(s)1−ψ(s)
s . (1.4)
The general form of the PDF of displacement for CTRW model may be derived from subordination approach as
p(x, t) =
∞
X
0
pn(x)χn(t).
Passing to the Laplace domain and Fourier space, we note that the Fourier
trans-1 Introduction
form ofPn(x) isn-th power of its characteristic function λn(k), substituting Eq.1.4 and calculating the sum one gets in Laplace-Fourier
P(k, s) = 1−ψ(s) s
1
1−ψ(s)λ(k). (1.5)
Substituting λ(k) andψ(s) as given functions in equation above and performing inverse Fourier and Laplace, one could in principle obtain the corresponding PDF of displacement in CTRW model. It should also be mentioned that substituting λ(k) andψ(s) possessing second and first moment, respectively, results in normal diffusion with Gaussian probability distribution function and substituting ψ(s) lacking the first moment results in anomalous diffusion with the strong deviation from Gaussian.
One important feature of the CTRW model which makes it easy to distinguish from other models isaging. Aging in fact, means at dependency of system’s evolution on the time when the observation starts. Suppose the observation starts at timeta
after the preparation of system. The MSD, for instance, is then given as hx2(t)i=hl2i(hn(ta+ ∆t)i − h(ta)i).
For those waiting time PDFs which the mean waiting time exist, we haven(t) =t/τ and the MSD has no dependency on ta, hx2(t)i ∝ ∆t/τ. For heavy-tailed waiting time PDF, however, we have n(t) ∝ tα so that hx2(t)i = hl2i((ta+ ∆t)α−tαa). It is now clear that the MDS in time ∆t explicitly depends on the time ta, in other words the systems ages. Other features of the system in the same way are dependent on the age of the process. In chapter 3, we will discuss the CTRW model in more details.
1.3.2 Random Walk in Disordered Systems
In the previous sections, it was assumed that diffusion takes place in homogeneous media. This means that whole space for the particle was allowed to be explored and from probabilistic point of view, particles have equal chances to move in all direc-tions, namely, an isotropic situation. In heterogeneous system, this is not the case and presence of large obstacles, such as macromolecules in living cells, prevents the diffusing element to access whole space and this imposes an anti-correlation of steps.
Studying such situation from mathematical point of view is not easy and in general, in these type of problems, proposing a closed formalism is not possible, except for some idealised situations. One approach for better understanding of diffusion in obstructed systems is to map the situation at hand to the problem of transport in a fractal systems. The self-similarity of the fractal systems allows mathematicians to use the re-normalization approach to obtain some analytical results for the transport process in the system. Note that this self-similarity in real cases is not perfect but still provides a good estimation of the situation.
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1.3 Anomalous Diffusion
Let us now define some features of the fractal systems which will be used in the next chapter for our analysis of FCS data. Analogous to the space dimension in Euclidean geometry, one defines the fractal dimension, df, for the fractal system which determines the growth of mass with size of structure, M(x) ∼xdf (?). The trajectory of random walker in the fractal substrate results also in another self-similar structure which its dimension can be attributed to a fractal dimension called walk dimension,dw. If the mass of newly created trajectory grow with the number steps, n, then one obtains the MSD of random walker to be (?)
hx2(n)i ∼n2/dw (1.6)
Assigning each step of random walker to an instant of time, one can simply replace, nandtin Eq.1.6. Note that exponent of anomalous diffusion is now given as, 2/dw, withdw = 2 leading to normal diffusion.
Derivation of PDF of particle’s displacement in fractal system is rather compli-cated and a general form is not available for arbitrary fractal structure. In principal, a long time simulation is required in order to estimate the form of PDF, see (?). Nev-ertheless, the probability of finding a random walker at the initial position at timet0, after time tis given asP(0, t)∼t−ds2 , which ds is the spectral dimension of fractal.
Spectral dimension can be considered as parameter determining the available sites for a random worker and may be interpreted as density of states in a fractal space (?). The connection between df,dw and ds was proposed by Alexander-Orbach (?)
as ds
2 = df
dw (1.7)
meaning that finding the walk and spectral dimension from a specific measure-ment, would lead to an estimation of fractal dimension of the substrate.
Percolation cluster
Percolation model has been continuously used to characterize many disordered sys-tems and extensively studied by mathematicians and physicists in simulation and theory (????). Often use of percolation cluster in order to mimic the situation of complex systems such as living cells does not mean that this is the only one but it is the well understood one which is often invoked for explanation of anomalous diffusion in crowded systems. Consider a cubic (square) lattice in three (two) di-mension, consisting of sites which are connected to nearest neighbours with bonds.
Assuming that the p fraction of these bonds are being removed, percolation cluster is thought as a limiting situation in whichpc= 0.592(0.401) fraction of these bonds are removed, where the lattice belongs to an incipient infinite cluster. Below this critical concentration, the system consists of a large cluster and above which there exist many finite clusters which the random walker can not percolate through the
1 Introduction
system (?).
Plethora of researches have shown that the percolation cluster is well described by a fractal (??). Thus diffusion on the percolation lattice may be described by the language of the fractals as follows: Above the critical point, pc, where there exist large clusters, the mean squared displacement starts to be anomalous and after some time, normal diffusion emerges. This is explained by the correlation or self-similarity length, ζ(p). For smaller length of the network compared to ζ(p), the self-similarity leads to anomalous diffusion, however, for larger sizes, the cluster turned to be homogeneous and diffusion is regular withdw = 2 in Eq.1.6. Therefore, one observes a crossover from anomalous to normal diffusion. Belowpc, small clusters are created. Therefor, random walker is trapped on one of these clusters and after experiencing anomalous diffusion in short times, MSD goes to a constant being square of correlation length at long times. Exactly at percolation threshold, the incipient infinite cluster is self-similar in all length scales and anomalous diffusion takes place withdw >2, with no crossover to normal diffusion.
This anti-correlation of displacement in percolation threshold is closely related to the anomalous diffusion modelled with GLE with coloured noise which will be the topic of following section.
1.3.3 Visco-elastic Systems
Diffusion of a particle in a visco-elastic system is another physical picture where the anomalous diffusion appears to be the case. In such picture, the diffusing particle is a part of complex system and its movement is a function of dynamic of the whole system. This in principal imposes a long term memory in the particle’s displace-ment which determines a concreted way for its movedisplace-ment (??). To mathematically approach this interpretation of anomalous diffusion, first we shortly mention the Langevin equation for normal diffusion and pass to the anomalous one.
Celebrated Lengevin equation (?) describes the Brownian motion of a spherical particle immersed in a solution with use of Newton laws for forces. It is assumed that the particle with mass m, experiences a random force f(t) (Gaussian white noise) and consequently the deterministic friction force−γx(t). Thus the trajectory˙ is given by the following stochastic differential equation
m¨x(t) =−γx(t) +˙ f(t) (1.8) where γ denotes the friction constant, directly connected to the viscosity of so-lution η and radius of particle a as γ = 6πηa. If the the system does not show a visco-elastic behaviour, ensemble average of the randomly fluctuating force (noise) is zero,hf(t)i = 0, and the noise is delta correlated, namely the random force is a white Gaussian noise.
10
1.3 Anomalous Diffusion
From the fluctuation-dissipation theorem, one gets the noise correlation function as:
hf(t).f0(t0)i= 2γKβT δ(t−t0)
withKβ andT being the Boltzmann constant and the temperature of the medium.
Calculating the velocity-velocity correlation function from the noise correlation func-tion and then using Green-Kubo relafunc-tion (?), one obtains the linear MSD as Eq.1.3 for Brownian particle with diffusion coefficient being KβT /γ.
Genrealized Langevin equation is essentially the modification of Eq.1.8, for the situation where the friction term contains an intrinsic memory of the environment Γ(t), say visco-elastic system, realized as an integral expression in Langevin equation as
mx(t) =¨ −γ Z t
0
Γ(t−t0) ˙x(t0)dt0+f(t). (1.9) Depending on whether the system is in equilibrium or not, the fluctuation-dissipation theorem applies or violates. In case of equilibration, the memory kernel and noise correlation function are connected ashf(t).f0(t0)i =KβT γΓ(t−t0) but, in essence, the connection between the noise, the friction term and the exponent of anoma-lous diffusion is connected to the setting of the system. The PDF of particle’s displacement remains Gaussian as long as the noise , say the fluctuation, is assumed to remain Gaussian and therefore the dynamic is totally defined by the MSD of particle. Note that, assuming the memory to be described by power-law kernel Γ(t)∝t−β, Eq.1.9 could be reformulated with the Caputo fractional derivative (?) and the equation is then referred as fractional Langevin equation (FLE). A good example of application of GLE for explanation of observed anomalous diffusion in single protein molecule may be the work by Kou, et al (?).
Fractional Brownian motion (fBm), rigorously introduced by Mandelbrot and Van Ness (?), is also a process that is closely related to the limiting case of GLE for long times. Rather than being a physical model, fBm is a mathematical model for conveniently describing the anomalous diffusion. Suppose that the position of particle obeys fBm,x(t) =BH(t) withH ∈(0,1) being the Hurst parameter,BH(t) is Gaussian distributed trajectory with zero mean and a persistent position-position correlation as
hBH(t)BH(t0)i=|t|2H +|t0|2H − |t−t0|2H/2. (1.10) fBm, in another words, takes the Brownian motion and incorporates a persistent correlation, which results in MSD of particle to be< x2(t)>∝t2H, whichH= 1/2,1 leads to normal and ballistic motion. In connection to the GLE in over-damped situation, fBm may be written as integration over Gaussian noise (FGN), f(t) in Eq.1.9, BH(t) = R0tf(t0)dt0. Again, the PDF of fBm process is Gaussian but in
1 Introduction
contrast to the Brownian motion it is not a Markov process and the whole statistic of process is not fully driven from the propagator.